Solving Dynamic Economic Dispatch Problems Facilitating Meta-Heuristic Technique

Solving Dynamic Economic Dispatch Problems Facilitating Meta-Heuristic Technique

K. C. Meher1

Department of Electrical Engineering Orissa Engineering College Bhubaneswar

R. K. Swain2

Department of Electrical Engineering Silicon Institute of Technology Bhubaneswar

1. K. Chanda3

   Department of Electrical Engineering Indian Institute of Engineering Science and Technology, Kolkata

   Abstract:-This paper yields glowworm swarm optimization technique to solve the dynamic economic load dispatch (DELD) problem for minimizing the operation cost in most economical manner. The glowworm swarm based algorithm is considered as meta-heuristic based evolutionary approaches where the agents communicate with each other via bioluminescent glowing, that enables them to explore the objective function more effectively. The effectiveness of the proposed method is demonstrated through five unit test systems. The results are compared with the other optimization technique reported in the literature. The fuel cost and other performance of the given approach has been quite impressive. It is shown that the GSO approach can provide the global or near global optimum solutions with a lesser computational time along with higher efficiency and robustness.

   KeywordsDynamic economic load dispatch (DELD), glowworm swarm optimization (GSO), luciferin, Valve – point loading effect, Ramp Rate Limits.

   find global optimal solution due to their non linear and non convex characteristics of generator input. Many stochastic search algorithm such as genetic algorithm(GA)[7,8], simulated annealing (SA) [9],evolutionary programming (EP) [10-13],particle swarm optimization(PSO) [14-16], differential algorithm DE [17-18] and clonal selection technique (CSA) [19] have been used in an effective manner to solve dynamic economic load dispatch problems. Recent researches have been directed towards the application of Glowworm swarm optimization (GSO) technique to solve DELD problem. The GSO is an efficient global search technique and may be used to find optimal or near optimal solutions to numerical and qualitative problems. It is easy to implement in most programming languages and has been proved to be quite effective and reasonably quick when applied to a diverse set of optimization problems.

   1. INTRODUCTION

      Dynamic economic load dispatch (DELD) plays a significant role in power system operation and control. It is a dynamic problem due to dynamic nature of power system and the large variation of load demand. This absolute problem is normally solved by discretization of the entire dispatch period into a no of small time intervals. The load is assumed to be constant and the system is considered to be in accurate steady state dynamic model which finds the best generation schedule for the generating units in real power system framework. The main objective of the dynamic economic dispatch is to minimizing the generation cost, subject to satisfy the physical and operational constraints. In traditional economic dispatch, the cost function is quadratic in nature. In practice, a

      In this paper, the GSO based DELD algorithm has addressed for the determination of global or near global optimum dispatch solution. In the proposed work, the operating limit constraints and valve point loading effects are fully incorporated. It has been shown that the algorithm is capable of finding the global or near global optimum solutions for large optimization problems.

      .

   2. PROBLEM FORMULATION

The main objective of the dynamic economic dispatch is to determine the outputs of all generating units to minimize the operating cost over a certain period of time under various physical and operational constraints. The formulation of DED problem has expressed as given below.

generating unit cannot exhibit a convex fuel cost function, so T N

a non-convex characteristics will observe owing to valve point effect. Mathematically, DED problem with valve point

Min F Fit Pit

t 1 i1

(1)

effect can be recognized as a non linear, non convex and large scale optimization problem with various complicated constraints, which finds the optimal result dispatch a new challenge. DED has been recognized as a more accurate problem than the traditional economic dispatch problem.

Many classical optimization techniques have been put on to solve the DED problems. The classical mathematical approaches include linear programming [1], nonlinear programming (NLP) [2], quadratic programming (QP) [3] , Lagrange relaxation(LR) [4,5],and Dynamic Programming (DP) [6]. However, most of these techniques are not able to

F : Total operating cost of all generating units over all dispatch periods.

T : Numbers of hour in the time horizon;

N : Number of generating units;

Fit(Pit): The fuel cost in terms of its real power output Pit at a time t

The fuel cost function with valve point effect of the thermal generating unit is expressed as the sum of a quadratics and sinusoidal-functions.

F (P ) a P 2 b P c | e (sin( f (P

P )) |

(2)

The number of luciferin level associated with glowworm i at

it it

i i i i i i

i i min i

time t. To update the current position of glowworm i , the

ai , bi

, ci ,

ei and

fi are constants of fuel cost function of

fitness value of the liciferin is given as follows:

ith unit.

li (t) (1 )li (t 1) J (xi (t))

(7)

1. Real power balance constraints

   Where

   : Luciferin decay constant whose value lies

   N

   i1

   (3)

   Pit

   • PDt

   • Plt

   0,

   t 1,2,T

   between (0,1)

   : Luciferin enhancement constant

   J : Objective function

   Where

   PDt

   is the total power demand during tth dispatch

   The individual glowworm i encodes the objective function

   period. Plt Total transmission loss during tth dispatch period

   J (xi (t) at current position

   xi (t) into a luciferin values

2. Real power operating limits

   li (t) and broadcast the same within its neighborhood. The set

   Pit min Pit Pit max , i 1,2,T;

   of neighborhood Ni (t) is chosen according to higher

   t 1,2,T

   (4)

   luciferin value within dynamic search space domain. The chosen numbers of glow in the local decision range is given

   Where

   Pi min and

   Pi max are the minimum and maximum real

   by the following equation

   '

   power outputs of ith generator respectively.

   Ni (t) j : x j (t) x (t) r ;l (t) l j (t)

   (8)

   i d i

3. Generator unit ramp rate limit

   Where,

   N (t) r ' ; r ' is the local updated decision range.

   i d d

   Pi(t1) Pit DRi ;

   i 1,n

   (5)

   The decision range is bounded by a circular sensor range

   r (0 r ' r ) , j N (t)

   s d s i

   Pit Pi(t 1) URi ;

   i 1,n

   (6)

   ith

   x (t) :Glowworm ith position at t iteration

   j

   l (t) :Glowworm ith luciferin at t iteration

   j

   URi and DRi are ramp up and ramp down rate limits of

   generator respectively and these are expressed in MW/h.

   The main objective of DELD problem is to determine the optimal schedule of output powers of online generating units with predicted power demands over a certain period of time to meet the power demand at minimum operating cost

   Each glowworm i selects a neighborhood glowworm j with probability Pij (t) .These movements enables the glowworm into a disjoint subgroup which finds the multiple optimal

   solutions to the objective function. The selection of the

   neighborhood glowworm by using probability distribution is given by

   III .OVERVIEW OF GLOWWORM SWARM OPTIMIZATION

   P (t)

   l j (t) li (t)

   (9)

   K.N. Krishnanad and D.Ghose [25] developed a glowworm ij

   swarm optimization (GSO) based on the behavior of glowworm (insects). The biological behaviors of the

   (lk

   kNi (t )

   (t) li (t))

   movement of individuals (e.g. ants, honeybee swarm, fish schools) within a group are the main observation in this algorithm. This algorithm consists of agents, which discover

   The updated luciferin movement is given by the following

   equation

   x j (t) xi (t)

   x

   i

   x (t 1) x (t) s

   (10)

   the search space and transmit the information regarding i fitness with respect to their correct position. The GSO algorithm searches the agents in a group of individuals similar

   j (t) xi (t)

   to the other heuristic based optimization methods. All individuals in a swarm approaches to the optimum or quasi- optimum through its randomly chosen direction of luciferin.In GSO algorithm, the search space composed N-dimensional agents called glowworm. Initialize the glowworm randomly in

   the search space. The position of the glowworm i at time t is

   Where s is called moving step size.

   In the last phase, the fitness of luciferin within a dynamic decision domain is upgraded in order to limit the range of communication and the updated local decision range is given by the following equation

   X i (t) (xi1 (t), xi 2 (t)xiN (t)) . The GSO algorithm has

   0, r ' (t)

   r ' (t 1) min r , max d

   s

   (11)

   described by the set of variables such as position vector d

   (n N )

   X i (t) , luciferin level li (t) and neighborhood range ri (t) .

   i i

   d

   r ' (t 1) : is the glowworm i' s local decision range at the

   (t 1) iteration

4. Decision of neighborhood

   The better the fitness value, the better the level of luciferin. Each glowworm finds all the glowworms which have the

   rs : The sensor range

   brighter luciferin level within the local dynamic decision

   range r ' . The decision range is bounded by a circular sensor

   ni : The neighborhood threshold value

   : Constant parameter

   1. IMPLEMENTATION OF GSO TO DYNAMIC ELD

      This section describes the implementation of GSO algorithm for solving dynamic economic load dispatch problems. This section deals the implementation of the equality and inequality constraints of DELD problems when modifying each individuals search space in the GSO algorithm. For T intervals in the generation scheduling periods, there are T dispatched for the N generating units. An array of decision variable vectors can be expressed as

      p11 p12 p1N

      P = p21 p22 p2 N

      it

      pT 1 pT 2 pTN

      Where Pit is the real power output of generator i at interval t

      1. Initialization of glowworm

         In the initialization procedure, the candidate solution of each individual (generating units power output) is randomly initialized within the feasible range in such a way that it should satisfy the constraint given by Eq. (4). A candidate is initialized as Pit ~ U ( Pit min, Pit max ) , Where U is the uniform distribution of the variables ranging in the interval of (Pit min, Pit max ) .

      2. Fitness function evaluation

         The fitness value of individual i is calculated the following equation (2). The number of population is equal to the number of fitness evaluation.

      3. Calculation of luciferin level

         Each glowworm carries a luminescent quantity called luciferin. The number of luciferin level associated with each glowworm i at times t . Put the value of objective function

         J (xi (t)) into the li (t ) in equation (7). The number of luciferin level is same as the number of glowworms.

         d

         range. The number of brighter glowworm is calculated by the following equation (8).

5. Updated local decision range

   At each iteration, the dynamic decision range is upgraded by the equation (9). The suitable value of is chosen such that

   how it effects the rate of change over the neighborhood range.

6. Glowworms velocity updated

   In this section, each glowworm carries their topical information which enables the glowworm to make their division into a number of subgroups. It gives the multiple local optimal solution of the given objective function. In order to select the proper neighbor, a probability distribution is chosen by the aforesaid equation (10)

7. Position updated of glowworms

The position of each glowworm is usually updated by the equation (11). The resultant position of individuals is being violated their operating limits and to keep the position as the individual within the boundary.

1. SIMULATION RESULTS AND DISCUSSION

   The proposed GSO algorithm has been applied to DELD problems by considering five- generating unit systems to investigate the effectiveness and robustness. The result obtained from proposed approach has been compared with EAPSO [20] and other previously developed techniques. The software has been written in MATLAB-7.5 language and executed on a 2.3-GHz Pentium IV personal computer with 512-MB RAM. For implementing the GSO technique in ELD problems, the maximum number of generation (iteration) of 5000 is taken for simulation study of the five unit test systems.

   Description of Case Study for the 5-unit Test Systems: Initially, the proposed GSO technique is applied on a small test system, consisting of five generating unit with a ramp rate limit and valve point effects The system data of this test have been taken from [20].Table I indicates the generator data of system. Table II yields the variation of load demand for 24 hours. The results obtained for this case study are given in table III, which shows that the GSO has better solution without violating any operating constraints The best fuel cost result obtained from proposed GSO and other optimization algorithms are compared in table IV. From table IV, it is clearly visible that the proposed method outperforms all other methods in terms of best fuel cost and simulation time. Fig 1.shows the variation of power generation with time and Fig

   2. shows the convergence behavior of the proposed technique.

   TABLE I

   Generator data for the 5-unit test system

   /tr>

   Units	ai	bi	ci	di	ei	Pi max	Pi min	DRi	URi
   1	25	2.0	0.0080	100	0.042	75	10	30	30
   2	60	1.8	0.0030	140	0.040	125	20	30	30
   3	100	2.1	0.0012	160	0.038	175	30	40	40
   4	120	2.0	0.0010	180	0.037	250	40	50	50
   5	40	1.8	0.0015	200	0.035	300	50	50	50

   TABLE II

   Load Variation for 24hours

   Load Demand(MW)
   Hour	Load	Hour	Load	Hour	Load	Hour	Load
   1	410	7	626	13	704	19	654
   2	435	8	654	14	690	20	704
   3	475	9	690	15	654	21	680
   4	530	10	704	16	580	22	605
   5	558	11	720	17	558	23	527
   6	608	12	740	18	608	24	463

   Fig.1.Power Generation of five unit systems with time

   TABLE III VI. CONCLUSION

   Best Solution of Proposed Method for five unit systems

   Hour	Unit 1 (MW)	Unit 2 (MW)	Unit 3 (MW)	Unit4 (MW)	Unit5 (MW)	Load (MW)
   1	17.2944	99.1647	30.1945	124.5679	139.7079	410.7399
   2	19.2021	99.7085	51.1479	126.0288	139.9210	435.8877
   3	10.6900	98.7549	71.6084	155.4974	139.8204	475.7750
   4	10.5866	98.7642	77.4776	204.7948	140.4001	531.0706
   5	10.3069	96.2009	108.1316	205.9634	139.6188	559.3281
   6	39.4206	106.5602	113.3428	210.7621	140.3704	609.9846
   7	68.2616	98.1121	112.8731	210.1602	139.0849	628.0831
   8	74.2339	101.0520	113.2912	213.1890	154.8241	655.9041
   9	70.3845	97.9287	112.3251	209.8157	202.0374	692.0525
   10	55.8620	98.8737	112.3839	209.7366	229.5768	706.1667
   11	72.4644	98.3169	112.5360	209.5947	229.5910	722.6415
   12	67.7412	98.7137	136.9351	209.7345	229.5732	743.0782
   13	56.0396	98.5365	112.3371	210.0573	229.4653	706.5003
   14	41.0388	98.7888	112.9186	209.9416	229.7016	691.9781
   15	34.2447	98.5367	112.7467	180.2379	230.1968	655.5823
   16	10.8238	97.3308	112.5992	131.5030	229.1279	581.2506
   17	11.0130	97.7050	97.7086	124.3342	228.4528	559.2045
   18	17.2650	98.5567	112.9927	150.5157	230.2600	609.4548
   19	17.1378	98.9317	112.3211	198.3132	229.4672	655.5492
   20	46.6632	101.2134	112.8768	210.1392	235.5337	706.1987
   21	31.8222	98.3838	113.0804	209.5952	229.4778	681.7967
   22	10.4036	98.3138	108.6966	160.1477	229.0916	606.5144
   23	10.7665	98.5560	112.1879	125.0940	181.7279	528.1315
   24	10.5248	82.8039	108.1656	124.9296	137.7758	464.0102

   A simple and an efficient optimization technique based on GSO is addressed for solving DELD problems, taking account of the valve point effect. By considering a five-unit test system, it is observed that the proposed algorithm has better convergence characteristics, robustness and computational efficiency as compared to other heuristic methods. It is also clear from the result obtained that by different trials, the proposed GSO technique can avoid the shorting of premature convergence of other optimization to obtain good quality results. When more complex fuel cost characteristics is considered, the solution quality and computational efficiency are significantly better than other methods. Because of these dominant characteristics, GSO becomes an important tool for solving more complex optimization problems in power system.

   5.1

   5

   4.9

   Cost $/h

   4.8

   4.7

   4.6

   4.5

   4.4

   4.3

   4

   x 10

   TABLE IV

   Comparison of Various Methods

   Methods	Minimum Cost($/h)	Simulation time(sec)
   MSL[24]	49216.81	1.4
   SA[21]	47356.00	351.98
   APSO[22]	44678	—
   SOA- SQP[23]	40701.42	—
   EAPSO[20]	43784	2.00
   GSO	43414.12	1.39

   5 unit System with GSO

   VII. REFERENCES

   1. R.Jabr, A.Coonick and B Corry. A homogeneous linear programming algorithm for the security constrained economic dispatch problem, IEEE Transaction s on Power System. Vol. 15, issue 3, pp 930-937;

      Aug.2000

   2. S.Takriti &B.Krasenbrink A decomposition approach for the fuel constrained economic power-dispatch problem, European journal of operational research .Vol. 112, issue 2, pp.460-466, 199

   3. L.Papageorgiou and E. Fraga. A mixed integer quadratic programming formulation for the economic dispatch of generators with prohibited operating zones, Electric Power System Research, vol. 77, issue 10, pp.1292-1296, 2007

   4. A.Keib, H. Ma, and J. Hart, Environmentally constrained economic dispatch using the Lagrangian relaxation method, IEEE Transaction son PowerSystem.vol.9, issue 4, pp. 1723-1727, 1994

   5. K. Hindi and M. Ghani, Dynamic economic dispatch for large-scale power systems: A Lagrangian relaxation approach, Electric Power system research, vol.13, issue1pp.51-56, 19

   6. D.Travers, and R Kaye, Dynamic dispatch by constructive dynamic programming, IEEE Transactions on Power System. vol.3, issue 1, pp.72-78, 1998

   7. S.Basker, P. Subbaraj, and M Rao, Hybrid real coded genetic algorithm solution to economic dispatch problem, Computers and Electrical Engineering. vol.29, issue 3,pp. 407-419, 2003

   8. C. Chiang, Improved genetic algorithm for power economic dispatch of units with value-point effects and multiple fuels, IEEE Transactions on Power System. vol. 20, issue 4, pp. 1690-1699, 2005.

   9. K. Wong, C. Fong, Simulated annealing based economic dispatch algorithm, IEEE proceedings on Generation, Transmission and Distribution. vol.140, issue 6, pp. 509-515, 1993

   10. T.Jayabarathi, K.Jayaprakash, and D.Jeyakumar, Evolutionary programming techniques for different kinds of economic dispatch problems, Electric Power System Research. vol.73, issue 2, pp. 169- 176, 2005

   11. T. Victoire, and A. Jeyakumar, A modified hybrid EP-SQP for dynamic economic dispatch with valve-point effect, International journal of Electrical Power and Energy Systems. vol. 27, issue 8, pp. 594-601, 2005

   12. T. Victoire, and A. Jeyakumar, Hybrid EP-SQP for dynamic economic

       0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

       Iterations

       Fig.2.Convergence behavior of proposed GSO

       dispatch with valve-point effect, Electric Power System Research,vol.71, issue 1, pp. 51-59, 2004

   13. P. Attaviriyanupap, H. Kita, E. Tanaka, J. Hasegawa, A hybrid EP and SQP for dynamic economic dispatch with non smooth fuel cost function, IEEE Trans. Power Syst.2002; vol.17, issue 2, pp.411-416, 2002.

   14. T.A.A. Victoire, and A.E. Jeyakumar, Deterministically guided PSO for dynamic dispatch considering valve-point effect, Electric Power Syst. Res. vol. 73, issue 3, pp. 313-322, 2005

   15. Victoire, T.A.A., and Jeyakumar,A.E.Reserved constrained dynamic dispatch of units with valve point effects. IEEE. Trans. Power Syst.2005; 20(3): 1273-1282

   16. Su A Yuan, Y Yuan. et al., An improved PSO for dynamic load dispatch of generators with valve-point effects. Energy. vol. 34, issue 1, pp. 67-74, 2009

   17. X Yuan, L Wang, and Y Yuan. et al, A modified differential evolution approach for dynamic economic dispatch with valve-point effects, Energy Conversion and Management, vol. 49, issue 12, pp. 3447-3453, 2008

   18. X Yuan, L Wang, and Y Yuan. et al, A hybrid differential evolution method for dynamic economic dispatch with valve-point effects, Expert system with Applications. vol. 36, issue 2, pp. 4042-4048, 2009

   19. R.K.Swain, A.K.Barisal, P.KHota and R.Chakrabarti, Short Term Hydrothermal Scheduling Using Clonal Selection Algorithm, International Journal in Electrical Power and Energy System. Vol. 33, issue 3, pp. 647-657, March. 2011

   20. T.Niknam, F.Golestaneh, Enhanced adaptive particle swarm optimization algorithm for dynamic economic dispatch of units considering valve- point effects and ramp rates, IET Gener.Transm.Distrib, 2012, Vol6, issue5,pp424-435.

   21. Panigrahi.C.K,Chattopadhyay.P.K,.Chakrabarty.R.N,,Basu.M simulated annealing technique for dynamic economic dispatch Elect power compt.syst,2006,34pp577-586

   22. Panigrahi, B,. Ravikumar, P.V,. Das,S,. Adaptive particle swarm optimization approach for static and dynamic for static and dynamic economic load dispatch. Energy conversion Mang., 2008,49,pp1407- 1422

   23. Sivasubramani, S. Swarup, K.S.: 'Hybrid SOA-SQP algorithm for dynamic economic dispatch with valve point effects Energy, 2010, 35.(12),pp5031-5036

   24. Hemamalini,S,.Simon,S.P .Dynamic economic dispatch using artificial using Maclaurin series based Lagrangian method Energy cover, Manage,2010,5,pp2212-2219.

   25. Krishnanand,K.N.D.Ghose,D.Glowworm swarm optimization: a new model for optimizing multimodal functions, Computational Intelligence Studies,1(1):93-119,2009